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Difficulty: HardQuadrilaterals and Polygons

A convex pentagon has three interior angles that are equal in measure. If the remaining two interior angles are supplementary to each other, what is the measure of one of the three equal angles?

  1. A
    60∘60^\circ
  2. 120∘120^\circAnswer
  3. C
    150∘150^\circ
  4. D
    180∘180^\circ

Answer

The measure of one of the three equal angles is 120∘120^\circ.
A pentagon has 5 sides, so the total sum of its interior angles is (5−2)×180∘=540∘(5 - 2) \times 180^\circ = 540^\circ. The problem states that three angles are equal (we can call each xx) and the other two are supplementary. Supplementary angles add up to 180∘180^\circ. Setting up the algebraic equation gives 3x+180∘=540∘3x + 180^\circ = 540^\circ. Subtracting 180∘180^\circ from both sides yields 3x=360∘3x = 360^\circ. Dividing by 3, we find that x=120∘x = 120^\circ.

Step-by-Step Solution

1
Calculate the total sum of the interior angles of a pentagon.
The sum is (5−2)×180∘=540∘(5 - 2) \times 180^\circ = 540^\circ.
The formula for the sum of interior angles of any convex polygon is (n−2)×180∘(n - 2) \times 180^\circ, where nn is the number of sides.
2
Set up an algebraic equation representing the sum of the five interior angles.
3x+180∘=540∘3x + 180^\circ = 540^\circ
Three angles are equal (let them be xx), and the remaining two angles are supplementary, meaning their combined sum is exactly 180∘180^\circ.
3
Solve the equation for xx.
3x=360∘⇒x=120∘3x = 360^\circ \Rightarrow x = 120^\circ
Subtracting 180∘180^\circ from both sides isolates the 3x3x term. Dividing both sides by 3 gives the value of a single equal angle.

Key Concept

The sum of interior angles of a polygon and the geometric property of supplementary angles.
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Topics

Quadrilaterals and Polygons

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